[avz]

Interestingly, if we go in the opposite direction we also loose something:

Real numbers (1-dimensional): No more algebraic closure [1]

though as you suggest we do gain a nice ordering where "nice" means "compatible with operations". BTW, you can always impose some ordering on all of these sets. It is the orderings' compatibility with arithmetic that we lose going from real numbers to complex numbers.

This can be generalized a little. Real numbers fail to be algebraically closed because some real polynomials lack real solutions, e.g. xx=-2. Complex numbers "fix" this problem since there every n degree polynomial has exactly n roots (counting multiplicity). In a sense, quaternions overdo this, e.g. xx=-2 has an infinite number of solutions (exercise for the reader). However, since quaternion multiplication is not commutative, polynomials as traditionally understood are ill-defined. For example, axx, xax and xxa are different degree two quaternion "polynomials". Ugh.

[1]: https://en.wikipedia.org/wiki/Algebraically_closed_field

[dhosek]

I used to teach the expansion of the concept of numbers from ℕ to ℂ via a fictionalized history of the numbers (most notably, expanding ℕ to ℤ before introducing ℚ, but historically, rational and even irrational numbers preceded zero and negative numbers in mathematical understanding. It also elides over the various subdivisions of all the different kinds of irrational numbers that exist in ℝ\ℚ which could be a whole class in itself, but this was a math for liberal arts majors class and I had to make sure not to have their eyes glaze over too much.